February 17, 1921] 



NATURE 



807 



sides of triangles in pairs implies that of corre- 

 sponding angles ; also many propositions about 

 areas, expressed in numerical form, could be 

 verified with some accuracy. As a result of these 

 successes the whole system was accepted. But 

 this could not have been a demonstration of the 

 truth of the system ; however many times a con- 

 gruence proposition is verified for particular pairs 

 of triangles, it will never be possible to prove it 

 true for the next pair without the further assump- 

 tion of some principle of empirical generalisation 

 that is not included among the postulates. 



Thus the grounds on which Euclid's results 

 have been generally adopted in practice are not 

 those of logical deduction from postulates known 

 to be true a priori, as was for centuries believed ; 

 they are largely based on empirical generalisa- 

 tion. Thus two sciences with different funda- 

 mental data, but with many formally similar pro- 

 positions, have grown up : the original name of 

 Euclidean geometry will here be retained for 

 Euclid's own development, while the experimental 

 science of measurement will be called mensura- 

 tion. The latter was not developed on its own 

 account, the reason being probably that it is a 

 physical science, and that so long as a theory 

 gives results in accordance with observation, 

 physicists in general show little disposition to 

 investigate the security of its foundations. 



All experimental science depends on some postu- 

 late or postulates that imply that empirical 

 generalisation is justifiable : no amount of experi- 

 ment will enable us to make any inference unless 

 we have some principle that enables us to 

 generalise the results. Theories involving such 

 a principle may be called extensive, while those 

 not involving one may be called intensive. The 

 latter include the whole of logic. Now no process 

 of generalisation is used in pure geometry ; every 

 proposition is proved immediately and with com- 

 plete certainty for every instance of its terms. 

 Thus all pure geometry, Euclidean or otherwise, 

 is intensive, while all physical sciences, including 

 mensuration, are extensive. Thus, in the first 

 f,'rpat subdivision of scientific knowledge, geo- 

 metry and mensuration fall on opposite sides, and 

 nothing but rorffusion can arise from any attempt 

 to treat them as identical. In geometry postu- 

 lates are made about any point, or any line; in 

 mensuration these are necessarily unverifiable, 

 for they would have to be tested for every possible 

 instance before they could be treated as postu- 

 I.ites. They can thus be obtained only by 

 generalisation from results that arc obtained by 

 experiment, and therefore cannot possibly be 

 primitive propositions. 



It may be remarked in passing that generalisa- 

 tion was condemned by traditional logic. The 

 fact that scientific men have not studied it.s valid- 

 ity for themselves is the chief reason for the 

 present chaotic condition of the theory of .scien- 

 tific knowledge. This important and basic prin- 

 ciple appears to involve necessarily the notions of 

 probability and combination of observations ; yet 

 in works purporting to be theories of scientific 

 NO. 2677, VOL. 106] 



knowledge these topics are habitually ignored 

 altogether or else relegated to the last chapter. 

 Much credit is due to Dr. N. R. Campbell for 

 placing the theory of probability in its proper 

 position in his recent " Elements of Physics." 

 The value of his treatment is injured by the adop- 

 tion of the Venn definition of probability, which 

 is logically unsound and scientifically inapplicable ; 

 but to accord probability its true status in scien- 

 tific knowledge is the first great advance in the 

 theory. 



This neglect of mensuration, while geometry 

 was making rapid progress, has led physicists to 

 give undue attention to the latter subject and an 

 undue physical status to its concepts, especially 

 to that of space. Metrical geometry has come to 

 be regarded as the theory of the measurement of 

 space ; but space has no status among the subject- 

 matter of mensuration. A physical measurement 

 of length is always of the form : " When the zero 

 mark on the (so-called rigid and straight) scale 

 is in contact with the particle A, and the edge of 

 the scale is in contact with the particle B, B lies 

 between the nth and {n + i)th divisions of the 

 scale." This involves no reference to space. 

 Even the notion of rigidity, which is often 

 regarded as spatial, is not so in practice. It is 

 an experimental fact that many bodies exist which 

 under ordinary treatment have the property that, 

 if the distance between two points of the one can 

 be made to include the distance between two points 

 of the other, in some configuration, then, however 

 the bodies are displaced, this remains true; and 

 our rigid scales are such bodies. This, with the 

 purely mechanical process of scale dividing, is 

 all that is needed to make measurement possible. 

 In actual physical calculation, again, we simply 

 use the measures themselves without any reference 

 to " space "; and the final result can be stated 

 wholly in terms of such measures. If the notion 

 of spa(-e is introduced at any stage of the investi- 

 gation, it necessarily eliminates itself before the 

 close. TBere are, of course, many so-called 

 measurements of length in physics, such as the 

 diameter of the earth and the distance of the sun, 

 which cannot be made by means of rigid scales; 

 these are not, in the strict sense, measured at 

 all, but inferred from measures of certain angles by 

 means of physical laws obtained from other ex- 

 periments. Time, as actually used in physics, is 

 on a similar footing to distance measures, and not 

 to space. 



The relations between the measured positions 

 of bodies at different times form the subject of 

 dynamics, which reduces to mensuration in a 

 special case. It is an extensive science, and 

 therefore epistemologically quite different from 

 any four-dimensional geometry, which must from 

 its nature be intensive. For this reason we con- 

 sider that the theory of gravitation must be 

 treated on extensive lines, and disagree with Prof. 

 Eddington's presentation of Einstein's theory as 

 a section of geometry. The principle of the 

 irrelevance of the mesh-system, in particular, does 

 not seem to be presented in its true light in hit 



